Mathematical objects through the lens of two different theoretical perspectives: APOS and OSA

Application of the Complementarities of Two Theories, APOS and OSA, for the Analysis of the University Students’ Understanding on the Graph of the Function and its Derivative

“…The starting point to search for connections between the OSA and APOS theory is the inquiry into the use of the notion of "mathematical object" in both theories Font et al, 2016). Two relevant aspects to address this issue are: (1) the interest that has had for mathematics education questions about the nature of mathematical objects, their various types, the processes of constitution and forms of participation in mathematical activity and (2) the fact that OSA and APOS are examples of a group of theories that use the term mathematical object as a relevant construct of its theoretical framework.…”

“…Two relevant aspects to address this issue are: (1) the interest that has had for mathematics education questions about the nature of mathematical objects, their various types, the processes of constitution and forms of participation in mathematical activity and (2) the fact that OSA and APOS are examples of a group of theories that use the term mathematical object as a relevant construct of its theoretical framework. To materialize this dialogue, in Font et al (2016) a Genetic Decomposition (GD) was developed from APOS for the concept of derivative that was later analyzed from the point of view of OSA, in order to unravel some aspects that are implicit in the mechanisms of APOS related to the emergence of objects, and thus understand better its complexity and the possible relation between mental constructions. Results of this networking showed some commonalities and some links between these theories and signaled the complementary nature of their constructs.…”

“…Recent research papers (Badillo et al, 2011;Drijvers, Godino, Font, & Trouche, 2013;Font, Trigueros, Badillo, & Rubio, 2016;Fuentealba, Sánchez-Matamoros, Badillo, & Trigueros, 2017;Mackie, & Court, 2002;Pino-Fan, Godino & Font, 2011;Pino-Fan, Font, Gordillo, Larios, & Breda, 2017;Robles, Telechea & Font, 2014;Tall, 2013;Tiwari, 2007) on learning and teaching of Calculus have revealed the complexity of the fundamental mathematical objects in this area of mathematics (derivative, integral, etc. ) and the need to teach different partial meanings of these objects and connect them to each other for their understanding.…”

“…To achieve this goal, in this paper the way of understanding APOS comprehension is complemented by the use of OSA theoretical tools to better characterize the mathematical activity performed by the student and to be able to infer their understanding of the problem. The use of tools of OSA is incorporated because we consider, according to Font et al (2016), that a networking of APOS and OSA provides tools to explain the difficulties associated with the learning of specific concepts.…”

Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative

“…The following GD was designed for teaching and learning the concept of the derivative based on the previous studies (Asiala et al, 1997;Font, Trigueros, Badillo, & Rubio, 2016) and our personal teaching insights giving emphasis to its graphical understanding: I) Definition of the derivative of a function at x using the tangent of its graph.…”